Theorem exmidontri2or 7389

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri2or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7368	. . 3 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		onelss 4452	. . . . . . . 8 |- ( y e. On -> ( x e. y -> x C_ y ) )
3	2	adantl 277	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> x C_ y ) )
4		orc 714	. . . . . . 7 |- ( x C_ y -> ( x C_ y \/ y C_ x ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( x C_ y \/ y C_ x ) ) )
6		eqimss 3255	. . . . . . . 8 |- ( x = y -> x C_ y )
7	6, 4	syl 14	. . . . . . 7 |- ( x = y -> ( x C_ y \/ y C_ x ) )
8	7	a1i 9	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( x C_ y \/ y C_ x ) ) )
9		onelss 4452	. . . . . . . 8 |- ( x e. On -> ( y e. x -> y C_ x ) )
10	9	adantr 276	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> y C_ x ) )
11		olc 713	. . . . . . 7 |- ( y C_ x -> ( x C_ y \/ y C_ x ) )
12	10, 11	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( x C_ y \/ y C_ x ) ) )
13	5, 8, 12	3jaod 1317	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( x C_ y \/ y C_ x ) ) )
14	13	ralimdva 2575	. . . 4 |- ( x e. On -> ( A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. y e. On ( x C_ y \/ y C_ x ) ) )
15	14	ralimia 2569	. . 3 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
16	1, 15	syl 14	. 2 |- (EXMID -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
17		ontri2orexmidim 4638	. . . 4 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID z = { (/) } )
18	17	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
19	18	exmid1dc 4260	. 2 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> EXMID )
20	16, 19	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   \/ w3o 980   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   (/)c0 3468   {csn 3643  EXMIDwem 4254   Oncon0 4428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-exmid 4255  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by:  onntri52  7390